Physics 1st Paper · Classes 11-12 · Chapter 1 · 1.12

Errors in Measurement.

You weigh a 50 g body three times and get 45, 45.1 and 45.0 g. The readings agree almost perfectly, and yet the balance is wrong. How? Because “close” can mean two things: close to each other, or close to the truth. This note first tells the kinds of error apart, then learns to measure them.

1. Five kinds of error

Every measurement carries some uncertainty. By where they come from, errors fall into five kinds: instrumental, observational (personal), random (irregular), systematic (regular) and least-count error.

When an instrument is not accurate and sensitive, the result is an instrumental error, of three kinds: zero error, backlash error and level error. In a vernier scale, screw gauge, slide calipers or spherometer, if the ‘0’ of the main scale does not coincide with the ‘0’ of the vernier or circular scale, the error is a zero error.

In instruments built on the nut-and-screw principle, turning the screw one way and then the other may not move it by equal amounts: that is backlash error. It is larger in old instruments, because the hole in the nut wears bigger or the screw wears loose. Level error arises when an instrument is not properly levelled, as in a balance, a deflection magnetometer or a tangent galvanometer; levelling screws and a spirit level remove it.

Observational or personal error comes from mistakes or carelessness in observing. It covers personal, end-mark, parallax, pointer-mark and environmental errors. The remedy is to take care and repeat the reading. When the line of sight changes, the object seems to shift position; the resulting error is parallax error. Error from wear of the instrument after long use is end error.

When repeated readings of the same quantity differ even with proper care, the error is a random (irregular) error; it can be positive or negative. The remedy is to take many readings and use the mean. When the result is always lower, or always higher, than the true value, the error is a systematic (regular) error. Examples: end error of a meter bridge, end error of a potentiometer, and zero error of a screw gauge.

The smallest amount an instrument can measure is its least count: 1 mm (0.1 cm) for a metre scale, 0.01 cm for slide calipers and 0.001 cm for a screw gauge. Least-count error can show up as random error and as systematic error. The mean of many readings comes closer to the true value.

2. Accurate and precise are not the same

The difference between the true value and the measured value is the absolute error. An error caused by the observer's carelessness is a gross error.

Accuracy says how close a measured value is to the true value: the closer, the more accurate. Precision says how close repeated readings are to one another: the smaller their spread, the higher the precision.

A balance that shows 45 g for a 50 g body is not accurate; another that shows 49.5 g is. Readings such as 48, 49.2, 49, 49.2, 49.6 and 49.5 g lie close together, and that shows good precision.

Two handles, two kinds of error. Systematic error shifts every reading one way; random error scatters them. Make all four cases yourself.

Below are four sets of readings of one 50 g body. Tap each set and check it. One question decides it: the distance from the true value, or the spread among the readings themselves?

3. From the mean to the error

If n measurements give x₁, x₂, …, xₙ, their mean is taken as the true value:

x̄ = (x₁ + x₂ + … + xₙ)/n

Now see how far each reading lies from the mean, ignoring the sign. The mean of those distances is the mean absolute error e. Here the absolute error of one reading means |xᵢ − x̄|, its distance from the mean, not the gap from the true value met above. The result is written x = x̄ ± e.

e = (|x₁ − x̄| + |x₂ − x̄| + … + |xₙ − x̄|)/n

The ratio of the mean absolute error to the true value is the relative or fractional error, e/x̄. Multiply it by 100 and you get the percentage error: (e/x̄) × 100%.

When the true value is known, the percentage error comes out directly:

|true value − obtained value| / true value × 100%

4. Deviation: mean and standard

The deviation of each reading is dᵢ = xᵢ − A, where A is the mean. Average the deviations without their signs and you get the mean deviation d, with the result x = A ± d:

d = (|d₁| + |d₂| + … + |dₙ|)/n

Square the deviations, average the squares, then take the root, and you have the standard deviation D, with the result x = A ± D:

D = √((d₁² + d₂² + … + dₙ²)/n) = √(Σd²/n)

A small example: for deviations +1, −1, +2, −2, Σd² = 1 + 1 + 4 + 4 = 10, so D = √(10/4) = √2.5 = 1.58.

Put the six steps in the right order.

5. Work the book's examples

Example 1.6(3): seven readings of a current, 5.6, 5.8, 5.2, 5.3, 5.5, 5.1 and 5.7 mA. The sum is 38.2, the mean 38.2/7 = 5.46 mA, the mean deviation 1.54/7 = 0.22 mA, and the standard deviation √(0.4172/7) = 0.244 mA. Now fill in the steps yourself.

One more: g is measured as 9.89 while the true value is 9.81. The percentage error is (9.89 − 9.81)/9.81 × 100 = 0.815%. If g comes out as 10, the error is (10 − 9.81)/9.81 × 100 = 1.94%. Timing 20 swings and dividing by 20, instead of timing one, spreads the same stopwatch error over 20 swings, so the percentage error drops.

In example 1.6(2) the book's numbers differ: true g = 9.80, measured 9.81. The percentage error is (9.80 − 9.81)/9.80 × 100 = −0.102%. Use the numbers the book prints. The minus sign tells you the measurement lies above the true value.

6. Check yourself

Three questions in the exam's style. Start with the first.

In the second, watch which error belongs to which kind.

In the last, match the formula to the working.

The whole note on seven cards.

Source: Physics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Physical World and Measurement”, section 1.12.